I did not discover this polyhedron, although I wish I had, for it has quite a clever design.

The page where I found it (poorly-translated English version, where it’s called the “Trick Johnson,” whatever that means) is athttp://www.geocities.jp/ikuro_kotaro/koramu/1053_g2.htm). I generally don’t repost much work by others here, but, for the “Trick Johnson,” I’m making an exception. By appearance, it’s a near-miss to the Johnson solids, based on combining characteristics of the dodecahedron, the snub cube, and the snub dodecahedron. It has chiral four-fold dihedral symmetry.

If you understand Japanese, I’m sure there’s a lot of interesting information at that linked page. If, on the other hand, you don’t, there’s still a good reason to follow that link: making fun of Google-Chrome’s built-in translator.

With some work, I was able to figure out how to make my second near-miss candidate from two posts ago, using Stella 4d (available here), but the results show it is a “near near miss,” not a near miss. Like the first one, the triangles are visibly irregular — and so are the green rectangles; there are also four edge lengths, the longest of which is ~11% longer than the shortest. This is not close enough to qualify as a near-miss.

Not long after I made the image above, a friend I shall simply call T. (until and unless I have his permission to publish his full name) e-mailed me his own versions he made, also using Stella. Here’s what they look like. Each can be enlarged with a click.

These are improved in the sense that the triangles (and squares, in the second one) are regular, but this was done at the expense of the pentagons. At the top and bottom of the figures, the edges where pentagons meet other pentagons are ~6.8% shorter than the other edges of each figure.

These last two are more likely to qualify for actual “near-miss” status — that has yet to be decided — but I need to make it clear than I did not discover them alone, but as part of a team. In my versions, after all, the flaws are more severe. Also, we do not yet know whether or not a different individual or team found these same polyhedra earlier, as often happens.

With help from friends on Facebook, I was able to figure out how to make the second of the near-miss candidates in the last post, using Stella 4d: Polyhedron Navigator, a program available here. This is quite helpful, for Stella has a “measurement mode” than lets me determine just how far off from regularity a given polyhedron is. This is what the “unbelted” polyhedron from the last post looks like, with the pentagons regular:

In this polyhedron, although the pentagons are regular, the triangles are scalene, with angles measuring ~55.35, ~60.81, and ~63.84 degrees. Of the three edge lengths needed for this, the longest is ~9.1% longer than the shortest, and the triangles are definitely non-regular — by visual inspection alone. It is possible to “tidy up” the triangles a bit, but only at the cost of making the pentagons visibly irregular. This is enough to make the call on the “unbelted” near-miss candidate from the last post — it’s a “near near miss,” not a true “near miss.”

All polyhedra in the last post, as it turns out, are related to another near-miss, the discovery of which I had nothing to do with. It has six pentagonal faces, and four which are quadrilaterals. This near-miss may be found here: http://www.mathcurve.com/polyedres/enneaedre/enneaedre.shtml.

[Note: see the next post, also, for more about these polyhedra.]

Yesterday, I played for the first time with GeoMag toys, which I recently purchased. I was quite surprised to have what I believe to be a near-miss to the Johnson solids appear before me, one I’ve never seen, within just a few minutes:

Here’s what it looks like, when viewed from two other angles.

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The faces of this three-fold dihedral polyedron are six pentagons, twelve triangles, and nine quadrilaterals. The fact that it has been proven that only 92 Johnson solids exist means that all of these faces cannot be regular. However, the irregularity is so small that I could not detect it in this model.

Next, I used Polydrons to build a net of this near-miss candidate.

What to do next was obvious: remove the “belt” of nine quadrilaterals, creating a net for a second near-miss candidate.

Having constructed this net, I then returned GeoMags to build a 3-d model of this second, “unbelted” near-miss candidate.

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I then wondered if I could make a third such solid by removal of the triangles, all of which appeared to be the lateral faces of pyramids.

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Could I remove them? Yes, and I did so. Did this create a third near-miss candidate? No. The resulting polyhedron, shown immediately above, is non-convex, and therefore cannot be a near-miss. The faces with dihedral angles greater than 180° are the triangle-pairs found where the pyramids were in the previous model.

With the “belted” and “unbelted” polyhedra before this non-convex non-candidate, the next step is to share them with other polyhedra enthusiasts, get their input regarding the question of whether these are genuine near-misses, and see if these polyhedra have already been found, unknown to me, by someone else.

[Update: please see the next two posts for more on these near-miss candidates.]

For detailed information on this newly-discovered polyhedron, which is near (or possibly in) the “fuzzy” border-zone between the “near-misses” (irregularities real, but not visually apparent) and “near-near-misses” (irregularities barely visible, but there they are) to the Johnson solids, please see the post immediately before this one. In this post, I simply want to introduce a new coloring-scheme for the chiral tetrated dodecahedron — one with three colors, rather than the four seen in the last post.

In the image above, the two colors of triangle are used to distinguish equilateral triangles (blue) from merely-isosceles triangles (yellow), with these yellow triangles all occurring in pairs, with their bases (slightly longer than their legs) touching, within each pair. This is the same coloring-scheme used for over a decade in most images of the (original and non-chiral) tetrated dodecahedron, such as the one below.

Both of these images were created using polyhedral-navigation software, Stella 4d, which is availablehere, both for purchase and as a free trial download.

[Later edit: I have now found out I was not the first person to find what I had thought, earlier today, was an original discovery. What I have simply named the chiral tetrated dodecahedron has been on the Internet, in German, since 2008, or possibly earlier, and may be seen here: http://3doro.de/polyeder/.]

The images above show a new near-miss (to the Johnson solids) candidate I just found using Stella 4d, software you can try here. Like the original tetrated dodecahedron (a recognized near-miss shown at left, below), making this polyhedron relies on splitting the Platonic dodecahedron into four three-pentagon panels, moving them apart, and filling the gaps with triangles. Unlike that polyhedron, though, this new near-miss candidate is chiral, as you can see by comparing the left- and right-handed versions, above. The image at the right, below, is the compound of these two enantiomers.

Next are shown nets for both the left- and right-handed versions of the chiral tetrated dodecahedron (on the right, top and bottom), along with the dual of this newly-discovered polyhedron (on the left). Like the rest of the images in this post, any of them may be enlarged with a click.

A key consideration when it is decided if the chiral tetrated dodecahedron will be accepted by the community of polyhedral enthusiasts as a near-miss (almost a Johnson solid), or will be relegated to the less-strict set of “near-near-misses,” will be measures of deviancy from regularity.The pentagons and green triangles are regular, with the same edge length. The blue and yellow triangles are isosceles, with their bases located where blue meets yellow. These bases are each ~9.8% longer than the other edges of the chiral tetrated dodecahedron. By comparison, the longer edges of the original tetrated dodecahedron, where one yellow isosceles triangle meets another, are ~7.0% longer than the other edges of that polyhedron. Also, in the original, the vertex angle of these isosceles triangles measures ~64.7°, while the corresponding figure is ~66.6° for the chiral tetrated dodecahedron.

[Later edit: I have now found out I was not the first person to find what I had thought, earlier today, was an original discovery. What I have simply named the chiral tetrated dodecahedron has been on the Internet, in German, since 2008, or possibly earlier, and may be seen here:http://3doro.de/polyeder/.]

In the polyhedron above, the octagons, hexagons, and triangles are regular. The only irregularities are found in the near-squares, which are actually isosceles trapezoids with three edges of equal length: the ones shared with the octagons and hexagons. The trapezoid-edges adjacent to the triangles, however, are ~15.89% longer than its other three edges. As a result, two of the interior angles of the trapezoids measure ~85.44º (the ones nearest the triangles), while the other two (adjacent to the shorter of the two trapezoid bases) measure ~94.56º. In a rotating model, it can be difficult to see the irregularities in these trapezoids. Were someone to build an actual physical model, however, the fact that they are not squares would be far more obvious.

In case someone would like to build such a model, here is a net you can use.

As you can see on this Wikipedia page, near-misses are not precisely defined — nor can they be, without such a definition (including something such as “no edge may be more than 10% longer than any other) being unjustifiably arbitrary. Instead, new near-miss candidates are discussed among members of the small community of polyhedral enthusiasts with an interest in near-misses, and are either admitted to the set of recognized near-misses, or not, based on consensus of opinion. This isn’t an entirely satisfactory system, but it’s the best we have, and may even be the best system possible.

The shortest definition for “near-miss Johnson solid” is simply “a polyhedron which is almost a Johnson solid.” Recently, a new (and even more informal) term has been created: the “near near-miss,” for polyhedra which are almost near-misses, but with deviations from regularity which are too large, by consensus of opinion, to be called near-misses. This polyhedron may well end up labeled a “near near-miss,” rather than a genuine near-miss.

Several questions remain at this point, and once I have found the answers, I will update this post to include them.

Is this close enough to being a Johnson solid to be called a near-miss, or merely a “near near-miss?”

Has this polyhedron already been found before? It looks quite familiar to me, and so it is entirely possible I have seen it before, and have simply forgotten when and where I saw it. On the other hand, this “I’ve seen it before” feeling may be caused by this polyhedron’s similarity to the great rhombcuboctahedron (also known as the truncated cuboctahedron, and a few other names), one of the Archimedean solids.

Does this polyhedron already have a name?

If unnamed at this time, what name would be suitable for it?

All the images in this post were created using Stella 4d, and I also used this software to obtain the numerical data given above. A free trial download of this program is available, and you can find it at http://www.software3d.com/Stella.php. Also, since it was mentioned above, I’ll close this post with a rotating image of the great rhombcuboctahedron. Perhaps a suitable name for the near-miss candidate above would be the “expanded great rhombcuboctahedron,” although it is entirely possible that a better name will be found.

Update #1: I now remember where I’ve seen this before: right here on my own blog! You can find that post here. I could delete this, as a duplicate post, but am choosing not to. One reason: the paths I took to create these two identical polyhedra were entirely different. Another reason is that this post includes information not included the first time around.

Update #2: This was already discussed among my circle of polyhedral enthusiasts. As I now recall, the irregularity in the quadrilaterals was agreed to be too large to call this a true “near-miss,” so, clearly, it’s a “near near-miss” instead.

As a proposed new “near-miss” to the Johnson solids, I created this polyhedron using Stella 4d, which can be found for purchase, or trial download, here. To make it, I started with a tetrahedron, augmented each face with icosidodecahedra, created the convex hull of the resulting cluster of polyhedra, and then used Stella‘s “try to make faces regular” function, which worked well. What you see is the result.

This polyhedron has no name as of yet (suggestions are welcome), but does have tetrahedral symmetry, and fifty faces. Of those faces, the eight blue triangles are regular, although the four dark blue triangles are ~2.3% larger by edge length, and ~4.6% larger by area, when compared to the four light blue triangles. The twelve yellow triangles are isosceles, with their bases (adjacent to the pink quadrilaterals) ~1.5% longer than their legs, which are each adjacent to one of the twelve red, regular pentagons. These yellow isosceles trapezoids have vertex angles measuring 61.0154º. The six pink quadrilaterals themselves are rectangles, but just barely, with their longer sides only ~0.3% longer than their shorter sides — the shorter sides being those adjacent to the green quadrilaterals.

The twelve green quadrilaterals are trapezoids, and are the most irregular of the faces in this near-miss candidate. These trapezoids have ~90.992º base angles next to the light blue triangles, and ~89.008º angles next to the pink triangles. Their shortest side is the base shared with light blue triangles. The legs of these trapezoids are ~2.3% longer than this short base, and the long base is ~3.5% longer than the short base.

If this has been found before, I don’t know about it — but, if you do, please let me know in a comment.

UPDATE: It turns out that this polyhedron has, in fact, been found before. It’s called the “tetrahedrally expanded tetrated dodecahedron,” and is the second polyhedron shown on this page. I still don’t know who discovered it, but at least I did gather more information about it — the statistics which appear above, as well as a method for constructing it with Stella.

This is the rhombic dodecahedron, the dual of the Archimedean cuboctahedron.

While the rhombic dodecahedron has 12 faces, there are many other polyhedra made entirely out of rhombi, and most of them have more than twelve faces. An example is the rhombic enneacontahedron, which has two face-types: sixty wide rhombi, and thirty narrow ones. It is one of several possible zonohedrified dodecahedra.

As the next figure shows, the wide rhombi of the rhombic enneacontahedron have exactly the same shape as the rhombic dodecahedron’s faces, so the two polyhedra can be stuck together (augmented) at those faces. These wide rhombi have diagonals with lengths in a ratio of one to the square root of two.

The next picture shows what happens if you take one central rhombic enneacontahedron, and augment all sixty of its wide faces with rhombic dodecahedra.

Since this polyhedral cluster in non-convex, it can be changed by creating its convex hull, which can the thought of as pulling a rubber sheet tightly around the entire polyhedron. Here’s the convex hull of the augmented polyhedron above.

The program I use to make these rotating images, Stella 4d (which you can try here), has a function called “try to make faces regular.” If applied to the convex hull above, this function leaves the triangles and pentagon regular, and makes the octagons regular as well. However, the rhombi become kites. The rectangles merely change, getting slightly longer, while rotating 90º, but they do remain rectangles.

After creating this last polyhedron, I started stellating it. After stellating it eight times, I obtained this polyhedron:

Once more, I applied the “try to make faces regular” function.

This polyhedron has five-valent vertices where the shorter edges of the kites meet. These are also the vertices of pentagonal pyramids which use kite-diagonals as base edges. By using faceting (the inverse operation of stellation), I next removed these pyramids, exposing their regular pentagonal faces.

In this polyhedron, all faces are regular, except for the red triangles, which are isosceles. (There are also triangles — the pink ones — which are regular.) Each of these isosceles triangles has ~63.2º base angles, and a ~53.6º vertex angle, with legs just under 11% longer than the base. This is a judgement call, for “near-miss” to the Johnson solids has not been precisely defined, but I see an ~11% edge-length difference as too great for this to be classified as a “near-miss,” even though I would love to claim discovery of another near-miss to the Johnson solids (if it even turns out I am the first one to find this polyhedron, which may not be the case). It is close to being a near-miss, though, so it belongs in the even-less-precisely defined group of polyhedra which are called, quite informally, “near near-misses.”

For the sake of comparison, here is a similar polyhedron (included in Stella 4d‘s enormous, built-in library of polyhedra) which is recognized as a near-miss to the Johnson solids. (I do not know the name of the person who discovered it, or I would include it here — I only know it wasn’t me.) It’s called the “half-truncated truncated icosahedron,” and its longest ledges are just over 7% longer than its shorter edges, with the non-regularity of faces also limited to isosceles triangles. However, this irregularity appears in all of the triangles in the polyhedron below — and in the “near near-miss above,” the irregularity only appeared in some of the triangular faces.

When Norman Johnson first found, and named, all the Johnson solids in the latter 1960s, he came across a number of “near-misses” — polyhedra which are almost Johnson solids. If you aren’t familiar with the Johnson solids, you can find a definition of them here. The “near-miss” which is most well-known features regular enneagons (nine-sided polygons):

This is the dual of the above polyhedron:

As with all polyhedra and their duals, a compound can be made of these two polyhedra, and here it is:

Finding this polyhedron interesting, I proceeded to use Stella 4d (polyhedron-manipulation software, available at http://www.software3d.com/Stella.php) to make its convex hull.

Here, then, is the dual of this convex hull:

Stella 4d has a “try to make faces regular” function, and I next used it on the polyhedron immediately above. If this function cannot work, though — because making the faces regular is mathematically impossible — one sometimes gets completely unexpected, and interesting, results. Such was the case here.

Next, I found the dual of this latest polyhedron.

The above polyhedron’s “wrinkled” appearance completely surprised me. The next thing I did to change it, once more, was to create this wrinkled polyhedron’s convex hull. A convex hull of a non-convex polyhedron is simply the smallest convex polyhedron which can contain the non-convex polyhedron, and this process often has interesting results.

Next, I created this latest polyhedron’s dual:

I then attempted “try to make faces regular” again, and, once more, had unexpected and interesting results:

The next step was to take the convex hull of this latest polyhedron. In the result, below, all of the faces are kites — two sets of twenty-four each.

I next stellated this kite-faced polyhedron 33 times, looking for an interesting result, and found this:

This looked like a compound to me, so I told Stella 4d to color it as a compound, if possible, and, sure enough, it worked.

The components of this compound looked like triakis tetrahedra to me. The triakis tetrahedron, shown below, is the dual of the truncated tetrahedron. However, I checked the angle measurement of a face, and the components of the above compound-dual are only close, but not quite, to being the same as the true triakis tetrahedron, which is shown below.

This seemed like a logical place to end my latest journey through the world of polyhedra, so I did.